The Turbulence Problem in Fluids Engineering and a log(Re) Solution

The existence and smoothness of solutions to the Navier–Stokes equations is a well-known mathematical statement of the turbulence problem. In the context of fluids engineering, however, the turbulence problem refers to the ability to predict turbulence both accurately and efficiently. This talk first explores what constitutes a viable solution to the turbulence problem from a computational perspective, invoking the concept of time complexity. In computer science, a solution is considered viable if its computational cost scales polynomially with the logarithm of problem complexity—i.e., log(n), where n denotes the problem size, such as the number of digits in integer factorization. For turbulence, problem complexity can be associated with the number of active scales in the flow, typically characterized by the Reynolds number (Re). Under this framework, a viable solution to the turbulence problem should exhibit a cost scaling of log(Re). Conventional scale-resolving approaches (e.g., DNS, LES) have computational costs that scale polynomially with Re, and thus do not qualify as viable solutions in this context. While Reynolds-averaged Navier–Stokes (RANS) methods, as we will show, achieve the desired log(Re) cost scaling, they lack the predictive accuracy of scale-resolving tools. We present a multi-fidelity framework that combines low-fidelity models, high-fidelity simulations, and machine learning to bridge this gap. The approach enables extrapolation of the accuracy of scale-resolving tools from low to high Reynolds numbers while maintaining a nominal computational cost scaling of log(Re). We demonstrate the effectiveness of this methodology using the canonical periodic hill flow case.